Hello,
To remove all norton files and registry information, download the Norton Removal Tool from the following link and follow the instructions given:
http://www.lavasoft.com/products/select_your_product.php?PHPSESSID=e011354f80a49b7a 5d43e08ca6f3ebd3
You can ignore Step 3 on reinstalling the norton product.
There are many free antivirus programs out there, and many would advise to avoid Norton. Personally, I've prefer using Avast!. I've used it on all of my computers without any problems. However, AVG is also good.
Depending on what Norton product you had, the Norton product can give additional protection on top of antivirus protection such as antispyware and firewall protection.
Windows as a software firewall built in which is probably enabled anyway, and for most users this is adequate enough. As for spyware/adware, if you don't have one already, you should install and use a spyware/adware scanner regularly.
Spybot Search & Destroy and
Lavasoft Ad-Aware 2007 Free are good free spyware/adware scanners to use.